Proof That Being a T4 Space is a Topological Property

Theorem

If a T4 spaceis homeomorphic to a spacethenis T4.

Proof

Ifis a T1 space homeomorphic to a spacethenis T1. Supposeis a normal space andis a homeomorphism. Letandbe disjoint closed subsets ofthen andare disjoint open sets of

Sinceis normal there are disjoint open setsandinsuch thatand

The setsandare open inandandhenceis normal and T4.